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Difficulty: Very hardReflection of Light at Plane and Curved Mirrors

If a light ray undergoes successive reflections from two plane mirrors inclined at an angle θ\theta to each other in a single plane, the total angle of deviation produced in the ray is independent of the initial angle of incidence at the first mirror.

Answer: Answer

Answer

The statement is TRUE. The total angle of deviation experienced by a ray after two successive reflections at inclined plane mirrors is δ=360∘−2θ\delta = 360^\circ - 2\theta, which depends only on the angle of inclination θ\theta and is independent of the initial angle of incidence.
The net angular deviation for a ray undergoing two successive reflections at plane mirrors inclined at angle θ\theta is δ=360∘−2θ\delta = 360^\circ - 2\theta. Since the initial angle of incidence i1i_1 cancels out during geometric summation, the total deviation is completely independent of the angle of incidence.

Step-by-Step Solution

1
Determine the deviation at the first mirror
δ1=180∘−2i1\delta_1 = 180^\circ - 2i_1
For a single reflection at a plane mirror with angle of incidence i1i_1, the angle of deviation is δ1=180∘−2i1\delta_1 = 180^\circ - 2i_1.
2
Express the angle of incidence at the second mirror in terms of inclination angle θ\theta
i2=θ−i1i_2 = \theta - i_1
From the geometric construction of the ray path inside the triangle formed by the two mirror surfaces, the interior angle relationship gives i1+i2=θi_1 + i_2 = \theta.
3
Calculate the total deviation after both reflections
δ=δ1+δ2=(180∘−2i1)+(180∘−2i2)=360∘−2(i1+i2)=360∘−2θ\delta = \delta_1 + \delta_2 = (180^\circ - 2i_1) + (180^\circ - 2i_2) = 360^\circ - 2(i_1 + i_2) = 360^\circ - 2\theta
Summing the individual deviations eliminates the variable i1i_1, demonstrating that the total deviation depends only on the inclination angle θ\theta.

Key Concept

Total Angle of Deviation for Inclined Plane Mirrors
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Topics

Reflection of Light at Plane and Curved Mirrors

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